Meta-transactions are blockchain transactions that are signed and submitted by users which are then executed by an external party to pay the gas fees on their behalf.
By nature, transactions on the Ethereum blockchain require a small payment called a gas fee in order to be processed and validated by the network. This leads to the cost paradox where new users are not able to send transactions without first acquiring ETH to pay for the gas fee.
With the meta-transaction model, the user sends a transaction with a few extra technical details (hidden from the user, of course) that gets routed to a secondary network rather than directly to a target smart contract. The secondary network will parse the transaction details to ensure the user's signature is valid before submitting it to the blockchain. A commonly used method is for an application provider to deploy a funded smart contract that can pay for the user's gas fee and an additional service reward that the secondary network receives for acting as a relayer.
This means that users are able to interact with a Web 3 application from accounts that don't hold any ETH, increasing the application's likelihood to gain traction by expanding the potential userbase to those who have no prior familiarity with blockchain technology.
Gas Station Network is a network of relayers that are able to accept transactions from user accounts without ETH, fill them with gas by signing them, and relay them to the blockchain to be executed. The relayers take a small fee to be compensated for their efforts.